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<H3>compute centro/atom command 
</H3>
<P><B>Syntax:</B>
</P>
<PRE>compute ID group-ID centro/atom lattice 
</PRE>
<UL><LI>ID, group-ID are documented in <A HREF = "compute.html">compute</A> command
<LI>centro/atom = style name of this compute command
<LI>lattice = <I>fcc</I> or <I>bcc</I> or N = # of neighbors per atom to include 
</UL>
<P><B>Examples:</B>
</P>
<PRE>compute 1 all centro/atom fcc 
</PRE>
<PRE>compute 1 all centro/atom 8 
</PRE>
<P><B>Description:</B>
</P>
<P>Define a computation that calculates the centro-symmetry parameter for
each atom in the group.  In solid-state systems the centro-symmetry
parameter is a useful measure of the local lattice disorder around an
atom and can be used to characterize whether the atom is part of a
perfect lattice, a local defect (e.g. a dislocation or stacking
fault), or at a surface.
</P>
<P>The value of the centro-symmetry parameter will be 0.0 for atoms not
in the specified compute group.
</P>
<P>This parameter is computed using the following formula from
<A HREF = "#Kelchner">(Kelchner)</A>
</P>
<CENTER><IMG SRC = "Eqs/centro_symmetry.jpg">
</CENTER>
<P>where the <I>N</I> nearest neighbors or each atom are identified and Ri and
Ri+N/2 are vectors from the central atom to a particular pair of
nearest neighbors.  There are N*(N-1)/2 possible neighbor pairs that
can contribute to this formula.  The quantity in the sum is computed
for each, and the N/2 smallest are used.  This will typically be for
pairs of atoms in symmetrically opposite positions with respect to the
central atom; hence the i+N/2 notation.
</P>
<P><I>N</I> is an input parameter, which should be set to correspond to the
number of nearest neighbors in the underlying lattice of atoms.  If
the keyword <I>fcc</I> or <I>bcc</I> is used, <I>N</I> is set to 12 and 8
respectively.  More generally, <I>N</I> can be set to a positive, even
integer.
</P>
<P>For an atom on a lattice site, surrounded by atoms on a perfect
lattice, the centro-symmetry parameter will be 0.  It will be near 0
for small thermal perturbations of a perfect lattice.  If a point
defect exists, the symmetry is broken, and the parameter will be a
larger positive value.  An atom at a surface will have a large
positive parameter.  If the atom does not have <I>N</I> neighbors (within
the potential cutoff), then its centro-symmetry parameter is set to
0.0.
</P>
<P>Only atoms within the cutoff of the pairwise neighbor list are
considered as possible neighbors.  Atoms not in the compute group are
included in the <I>N</I> neighbors used in this calculation.
</P>
<P>The neighbor list needed to compute this quantity is constructed each
time the calculation is performed (e.g. each time a snapshot of atoms
is dumped).  Thus it can be inefficient to compute/dump this quantity
too frequently or to have multiple compute/dump commands, each with a
<I>centro/atom</I> style.
</P>
<P><B>Output info:</B>
</P>
<P>This compute calculates a per-atom vector, which can be accessed by
any command that uses per-atom values from a compute as input.  See
<A HREF = "Section_howto.html#howto_15">Section_howto 15</A> for an overview of
LAMMPS output options.
</P>
<P>The per-atom vector values are unitless values >= 0.0.  Their
magnitude depends on the lattice style due to the number of
contibuting neighbor pairs in the summation in the formula above.  And
it depends on the local defects surrounding the central atom, as
described above.
</P>
<P>Here are typical centro-symmetry values, from a a nanoindentation
simulation into gold (FCC).  These were provided by Jon Zimmerman
(Sandia):
</P>
<PRE>Bulk lattice = 0
Dislocation core ~ 1.0 (0.5 to 1.25)
Stacking faults ~ 5.0 (4.0 to 6.0)
Free surface ~ 23.0 
</PRE>
<P>These values are *not* normalized by the square of the lattice
parameter.  If they were, normalized values would be:
</P>
<PRE>Bulk lattice = 0
Dislocation core ~ 0.06 (0.03 to 0.075)
Stacking faults ~ 0.3 (0.24 to 0.36)
Free surface ~ 1.38 
</PRE>
<P>For BCC materials, the values for dislocation cores and free surfaces
would be somewhat different, due to their being only 8 neighbors instead
of 12.
</P>
<P><B>Restrictions:</B> none
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "compute_cna_atom.html">compute cna/atom</A>
</P>
<P><B>Default:</B> none
</P>
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<A NAME = "Kelchner"></A>

<P><B>(Kelchner)</B> Kelchner, Plimpton, Hamilton, Phys Rev B, 58, 11085 (1998).
</P>
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